Raievska I. Finite nearrings with identity and local nearrings with some restrictions on additive groups

Українська версія

Thesis for the degree of Doctor of Science (DSc)

State registration number

0526U000221

Thesis Registration Form

0526U000221.pdf

Applicant for

Specialization

  • 01.01.06 - Алгебра і теорія чисел

26-08-2026

Specialized Academic Board

Д 26.206.03

Institute of Mathematics of the National Academy of Sciences of Ukraine

Essay

The thesis is devoted to the study of nearrings with identity and local nearrings, and their construction using the computer algebra system GAP. Chapter 1 is devoted to a review of known results on finite nearrings with identity. It is obvious that every associative ring is a nearring, and every group is the additive group of a nearring, but not necessarily of a nearring with identity. The question of which groups can be additive groups of nearrings with identity has been studied since the late 1960s. The study of local nearrings was initiated by Maxson, who determined a number of their basic properties and, in particular, proved that the additive group of a zero-symmetric local nearring is a p-group. This author described all non-isomorphic zero-symmetric local nearrings with a non-cyclic additive group of order p^2 that are not nearfields. Note that it remains an open question which non-abelian p-groups can be additive groups of local nearrings. Chapter 2 presents the main definitions and results concerning nearrings with identity and groups of order p^3. Necessary conditions for the existence of nearrings with identity on such groups are obtained. Chapter 3 is devoted to the study of local nearrings of order p^3 and gives the definition of local nearrings and their basic properties. Examples of local nearrings with an elementary abelian additive group of order p^3 are constructed. Local nearrings on non-abelian groups of order p^3 are studied. Necessary and sufficient conditions for the existence of local nearrings whose additive groups are isomorphic to G_1, G_2 and G_3, respectively, are obtained. Chapter 4 studies nearrings with identity of order p^4. A classification of groups of order p^4 is given. Necessary conditions for the existence of nearrings with identity on groups of nilpotency class 2 and nilpotency class 3, respectively, are obtained. Chapter 5 describes local nearrings of order p^4. Subsection 1 is devoted to the study of local nearrings on groups of order p^4 and nilpotency class 2. Namely, it is shown that for odd p out of 6 such groups 4 are additive groups of local nearrings. Some examples of such nearrings are explicitly constructed. In subsection 2 groups of nilpotency class 3 of order p^4, which are additive groups of local nearrings, are studied. Chapter 6 studies nearrings with identity on some non-Abelian groups. All possible types of Miller–Moreno groups, which are additive groups of nearrings with identity, are described. It is shown that there are no nearrings with identity whose additive group is isomorphic to the Schmidt group. Direct products of metacyclic Miller–Moreno p-groups and cyclic p-groups as additive groups of nearrings with identity and local nearrings are studied. Moreover, multiplication formulas for such nearrings are defined and the functions that define them are described. Direct products of non-metacyclic Miller–Moreno p-groups and cyclic p-groups as additive groups of nearrings with identity and local nearrings are studied. Chapter 7 is devoted to the application of the GAP system to the study and classification of local nearrings. It is studied the construction nearrings with identity and classification of local nearrings. It is described some functions from the LocalNR package, namely, UnitsOfNearRing, IsLocalNearRing, IsLocalRing, NearRingNonUnits and NonUnitsAsAdditiveSubgroup. The classification of local nearrings of small orders, namely, local nearrings of order 32 and 128, is provided.

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Research papers

Sysak Ya., Raievska I., Raievska M. Construction of local nearrings using GAP. Algebra Discrete Math., 2025, 40, № 1, 133–144. https://doi.org/10.12958/adm2427

Raievska I., Raievska M., Sysak Ya. LocalNR package and some of its applications. Algebra Discrete Math., 2025, 40, № 2, 215–225. https://doi.org/10.12958/adm2449

Раєвська І., Раєвська М. Про ендоциклічні 2-породжені групи порядку 256 та експоненти 16. Буковинський матем. журнал, 2025, 13, № 2, 161–169. https://bmj.chnu.edu.ua/media/gbdhv1v1/161-169_raievska_raievska_cor.pdf

Раєвська І. Про прямі добутки неметациклічних p-груп Міллера–Морено та циклічних p-груп як адитивні групи локальних майже-кілець. Науковий вісник Ужгородського університету. Серія «Математика і інформатика», 2025, 46, № 1, 79–88. https://doi.org/10.24144/2616-7700.2025.46(1).79-88

Raievska I., Raievska M. Groups of the nilpotency class 3 of order p^4 as additive groups of local nearrings. Carpathian Math. Publ., 2025, 17, № 1, 292–301. https://doi.org/10.15330/cmp.17.1.292-301

Raievska I., Raievska M. Local nearrings, their structure, and the GAP system. Ukrainian Math. J., 2025, 76, № 11, 1831–1848, https://doi.org/10.1007/s11253-025-02426-y; translation of Ukrain. Mat. Zh., 2024, 76, № 11, 1629–1644

Raievska I., Raievska M. Local nearrings and endocyclic groups of small order. Scientific Bulletin of Uzhhorod University. Series of Mathematics and Informatics, 2025, 47, № 2, 65–71. https://doi.org/10.24144/2616-7700.2025.47(2).65-71

Raievska I. On direct products of metacyclic Miller–Moreno p-groups and cyclic p-groups as additive groups of local nearrings. Прикл. проблеми мех. і мат., 2024, 22, 123–130

Raievska I., Raievska M. Groups of order p^4as additive groups of local nearrings. Ukrainian Math. J., 2024, 76, № 6, 1005–1024, https://doi.org/10.1007/s11253-024-02369-w; translation of Ukrain. Mat. Zh., 2024, 76, № 6, 890–906

Raievska I., Raievska M. Groups of nilpotency class 2 of order p^4 as additive groups of local nearrings. Algebra Discrete Math., 2024, 38, № 1, 93—114. http://dx.doi.org/10.12958/adm2269

Raievska I. Yu. Local nearrings with additive groups of order 128, Науковий вісник Ужгородського університету. Серія ``Математика і інформатика'', 2024, 44, № 1, 46–50. https://doi.org/10.24144/2616-7700.2024.45(2).110-114

Raievska I., Raievska M. Lower bounds for the number of local nearrings on groups of order p^3. Math. Commun., 2024, 29, № 2, 177–191. https://www.mathos.unios.hr/mc/index.php/mc/issue/view/35

Раєвська І. Ю., Раєвська М. Ю. Локальні майже-кільця на елементарних абелевих групах порядку p^3. Науковий вісник Ужгородського університету. Серія ``Математика і інформатика'', 2021, 38, № 1, 85–93. https://doi.org/10.24144/2616-7700.2021.38(1).85-93

Raievska I. Yu., Raievska M. Yu. Local near-rings with a multiplicative Schmidt group. Ukr. Math. J., 2020, 71, №10, 1643–1649. https://doi.org/10.1007/s11253-020-01737-6; перекладено з Ukr. Mat. Zh., 2019, 71, № 10, 1435–1440. (in Ukrainian). http://umj.imath.kiev.ua/index.php/umj/article/view/1526/509

Raievska I. Yu., Raievska M. Yu. Finite nearrings with identity on Miller–Moreno groups. Mat. Stud., 2014, 42, № 1, 15–20. http://matstud.org.ua/texts/2014/42_1/15-20.pdf

Раєвська І. Ю., Раєвська М. Ю. Скінченні локальні майже-кiльця. Могилянський математичний журнал, 2018, 1, 38–48. http://mmj.ukma.edu.ua/article/view/152611/151700

Raievska I., Raievska M., Sysak Ya. DatabaseEndom32: (v1.0.2) [Data set] (2024). Zenodo. https://doi.org/10.5281/zenodo.10820301

Raievska I., Raievska M., Sysak Ya. DatabaseEndom128: (v0.2) [Data set] (2022). Zenodo. https://doi.org/10.5281/zenodo.7225377

Raievska I., Raievska M., Sysak Ya. DatabaseEndom625: (v0.2) [Data set] (2023). Zenodo. https://doi.org/10.5281/zenodo.7613145

Raievska I., Raievska M., Sysak Ya. LocalNR, Package of local nearrings, Version 1.0.4 (2024) (GAP package). https://gap-packages.github.io/LocalNR/

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